Mathematical Sciences: Dynamics and Patterns in Reaction Diffusion Equations

数学科学：反应扩散方程的动力学和模式

• 批准号: 9002349
• 负责人: David Terman
• 金额: $ 5.23万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-07-01 至 1993-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002349&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Dynamics; Patterns; Reaction

Reaction diffusion equations may give rise to a variety of pattern like solutions. These include traveling wave solutions, stable steady solutions, cellular patterns, oscillatory solutions, and even solutions which behave in a chaotic manner. All of the systems which the principle investigator will consider arise as a mathematical model of a biological or chemical system. This project will consider models for combusion, chemical reactions on catalytic surfaces, phase transitions, and bursting in excitable membranes. The primary goals of this research project are to investigate through which mathematical mechanisms these patterns may arise, analyse the stability properties of each pattern, and understand what physical significance each pattern has for the underlying physical problem. The problems to be addressed in this project raise a number of fundamental mathematical issues. Of particular interest are patterns arising in more than one space dimension. Much of the analysis will be concerned with how higher dimensional patterns bifurcate from basically one dimensional objects such as planar wave solutions and radially symmetric solutions. Simple examples demonstrate that the structure of the bifurcating patterns may be incredibly complex.

反应扩散方程可能会产生各种类似图案的解。这些解包括行波解、稳定稳定解、细胞模式、振荡解，甚至是行为混乱的解。首席研究者将考虑的所有系统都是作为生物或化学系统的数学模型出现的。这个项目将考虑燃烧、催化表面的化学反应、相变和可激发膜中的破裂的模型。这个研究项目的主要目标是调查这些模式可能通过哪些数学机制产生，分析每种模式的稳定性属性，并了解每种模式对潜在的物理问题具有什么物理意义。这个项目中要解决的问题提出了一些基本的数学问题。特别令人感兴趣的是出现在不止一个空间维度上的图案。大部分分析将关注高维图案如何从基本上一维的对象中分叉出来，例如平面波解和径向对称解。简单的例子表明，分叉模式的结构可能非常复杂。